Understanding the Small World Effect – Part I

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It appears that the seemingly random connections that cause us to declare “it’s a small world” can be used to explain everything from globalization to the epidemic spread of disease.

After an initial outbreak in Guangdong province of southern China in November 2002, the SARS virus spread via a Chinese academic who arrived in Hong Kong in February 2003. Then the Small World Effect kicked in, as those he infected travelled by air to Taiwan, Vietnam, Singapore and Canada – triggering local epidemics in these widely-separated places within days of each other. They, in turn, led to further outbreaks in places as far apart as Europe and Australia. Rapid identification of new cases and isolation of those infected people halted the epidemic within a few months, but not before 8000 had contracted the disease, of whom almost 1 in 10 died.

In February 2003, a Chinese academic arrived in Hong Kong, checked into his hotel and unwittingly began spreading a deadly virus around the world. He had flown in from southern China, where months earlier a strange new form of flu had emerged. Within days, he had been taken into hospital with similar symptoms, but not before infecting over a dozen other people, who then spread the virus to Taiwan, Singapore, Vietnam and Canada. By mid-March, the Severe Acute Respiratory Syndrome (SARS) virus had triggered a global health alert – and become a chilling example of what scientists call the Small World Effect.

As a social phenomenon, it’s familiar enough. Talking with a stranger at a party, we discover a friend or colleague in common. We respond with a smile, exclaim “Well, it’s a small world!”, and think no more about it. And yet, of course, the world isn’t small at all: with over 6.7 billion people, it’s huge. Somehow the Small World Effect short-circuits all that, bringing us all into surprising proximity to each other. In the case of the SARS virus, it has potentially fatal consequences. Understanding the Small World Effect is now a major area of research in such fields as economics, medicine and marketing. Perhaps the biggest surprise is that it has taken scientists so long to take the effect seriously. For while it has probably been noticed at social gatherings for centuries, it was only in the 1950s that researchers began to probe its roots.

At the University of Chicago, mathematician turned social scientist Anatol Rapoport and colleagues began by creating an artificial society and thinking of it as a ‘network’ of individual people, each with random links to others. Some links were short, connecting up people into close knit communities while others were relatively long. Rapoport and his colleagues were intrigued to find that the random nature of the links made a big difference to the structure of their artificial society. If the ties were made even just a little less random, the society tended to fragment into isolated communities, with no links with people elsewhere. It was an early hint of the crucial importance of random links in turning a huge population into a ‘small world’. Imagine a society that consists of a million people, each of whom has links only with 10 people living nearby.

If a rumour breaks out, it would take tens of thousands of steps for it to spread throughout the society as it plods round from one ‘clique’ to another. Now imagine the same society where, again, everyone knows 10 people but this time randomly spread across the society. The rumour spreads far more quickly, popping up at random anywhere. After each re-telling, the number of people knowing the rumour grows by a factor of 10 – first to 100, then 1000 and so on. After just six re-tellings, everyone will have heard the rumour, thanks to the random links. Which is all very impressive, except that the real world isn’t like that. We have ties that are neither totally random nor completely parochial, but a mixture of both. So why do we so often discover that we too live in a ‘small world’? In 1959 two Hungarian mathematicians, Paul Erdös and Alfred Rényi, showed how just a few random links can make a big difference. For example, in a crowd of 100 people, it’s possible to form ties between virtually everyone by forming random links between just a dozen or so. Achieving the same systematically, on the other hand, demands a staggering 4950 links. While Erdös and Rényi’s formula showed the power of random links, it took an ingenious experiment performed in 1967 to prove that we really do live in a ‘small world’.

Stanley Milgram, a young sociology professor at Harvard University, wanted to gauge the typical size of our social networks – how many people we have as friends, or friends of friends, and so on. To find out, he posted packages to 100 people in Nebraska and Kansas, asking them to post them on to a ‘target’ person in Massachusetts. Which sounds simple enough, except the recipients weren’t told where the target person lived, only their name, occupation and a few other personal details. Milgram asked the recipients to post the packet to anyone they knew on first-name terms who might have a better chance of being able to deliver the packet.

Post haste the outcome was stunning. The packets typically reached the target after just five re-postings. A few years later, Milgram repeated his experiment, with similar results. It seemed that everyone in the USA could be reached via just five re-postings. The implications were even more amazing. If just five posting were enough to reach anyone in a country of over 200 million, it seemed people typically knew around 50 others well enough to post the letter on. And with each of those 200 million knowing another 50 people that meant that just one more posting would be enough to reach 10 billion people – more than Earth’s entire population. Milgram’s finding was reaffirmed by researchers at Microsoft this year, who analysed patterns of instant messages passed around the world during one month in 2006 and found that each of us is typically linked to everyone else by no more than  seven intermediaries. Milgram’s discovery became an urban legend, and even gave its name to a play, Six Degrees of Separation by American playwright John Guare, in which one of the characters says: “Everybody on this planet is separated by only six other people. Six degrees of separation. It’s a profound thought.”

We will continue this very interesting article written by Robert Matthews in our next post. Robert is a science journalist and Visiting Reader in Science at Aston University, UK.

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